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Theorem al0ssb 5309
Description: The empty set is the unique class which is a subclass of any set. (Contributed by AV, 24-Aug-2022.)
Assertion
Ref Expression
al0ssb (∀𝑦 𝑋𝑦𝑋 = ∅)
Distinct variable group:   𝑦,𝑋

Proof of Theorem al0ssb
StepHypRef Expression
1 0ex 5308 . . 3 ∅ ∈ V
2 sseq2 4009 . . . 4 (𝑦 = ∅ → (𝑋𝑦𝑋 ⊆ ∅))
3 ss0b 4398 . . . 4 (𝑋 ⊆ ∅ ↔ 𝑋 = ∅)
42, 3bitrdi 287 . . 3 (𝑦 = ∅ → (𝑋𝑦𝑋 = ∅))
51, 4spcv 3596 . 2 (∀𝑦 𝑋𝑦𝑋 = ∅)
6 0ss 4397 . . . 4 ∅ ⊆ 𝑦
76ax-gen 1798 . . 3 𝑦∅ ⊆ 𝑦
8 sseq1 4008 . . . 4 (𝑋 = ∅ → (𝑋𝑦 ↔ ∅ ⊆ 𝑦))
98albidv 1924 . . 3 (𝑋 = ∅ → (∀𝑦 𝑋𝑦 ↔ ∀𝑦∅ ⊆ 𝑦))
107, 9mpbiri 258 . 2 (𝑋 = ∅ → ∀𝑦 𝑋𝑦)
115, 10impbii 208 1 (∀𝑦 𝑋𝑦𝑋 = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wal 1540   = wceq 1542  wss 3949  c0 4323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-dif 3952  df-in 3956  df-ss 3966  df-nul 4324
This theorem is referenced by:  iota0def  45748  aiota0def  45804
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